ARTICLE

DOI: 10.1038/s42003-018-0197-1 OPEN Slow recovery from a disease epidemic in the spotted hyena, a keystone social carnivore

Sarah Benhaiem 1, Lucile Marescot 1,2, Marion L. East 1, Stephanie Kramer-Schadt 1,3, Olivier Gimenez 2, Jean-Dominique Lebreton 2 & Heribert Hofer 1,4,5 1234567890():,;

Predicting the impact of disease epidemics on wildlife populations is one of the twenty-first century’s main conservation challenges. The long-term demographic responses of wildlife populations to epidemics and the life history and social traits modulating these responses are generally unknown, particularly for K-selected social species. Here we develop a stage- structured matrix population model to provide a long-term projection of demographic responses by a keystone social predator, the spotted hyena, to a virulent epidemic of canine distemper virus (CDV) in the Serengeti ecosystem in 1993/1994 and predict the recovery time for the population following the epidemic. Using two decades of longitudinal data from 625 known hyenas, we demonstrate that although the reduction in population size was moderate, i.e., the population showed high ecological ‘resistance’ to the novel CDV genotype present, recovery was slow. Interestingly, high-ranking females accelerated the population’s recovery, thereby lessening the impact of the epidemic on the population.

1 Department of Ecological Dynamics, Leibniz Institute for Zoo and Wildlife Research, Alfred-Kowalke-Strasse 17, D-10315 Berlin, Germany. 2 CEFE, CNRS, University Montpellier, University Paul Valéry Montpellier 3, EPHE, IRD, Montpellier 34090, France. 3 Department of Ecology, Technische Universität Berlin, Rothenburgstr. 12, 12165 Berlin, Germany. 4 Department of Veterinary Medicine, Freie Universität Berlin, Oertzenweg 19b, Berlin 14163, Germany. 5 Department of Biology, Chemistry, Pharmacy, Freie Universität Berlin, Takustr. 3, Berlin 14195, Germany. These authors contributed equally: Sarah Benhaiem, Lucile Marescot. These authors jointly supervised this work: Jean-Dominique Lebreton, Heribert Hofer. Correspondence and requests for materials should be addressed to S.B. (email: [email protected])

COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio 1 ARTICLE COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1

pidemics responsible for a decline in keystone species can performance, social status and infection state drive both R0 and Ealter ecosystem dynamics and diminish biodiversity by the population response to a major epidemic. increasing the chance of extirpation of host populations, 1–4 and possibly the extinction of species . Human activities rapidly Results expand the geographical range and host species spectrum of Demographic, social and infection parameter estimates. Each pathogens, and epidemics caused by exotic pathogens in unex- 5 year, each female was assigned a demographic (cub, subadult, pected hosts are increasing . Viruses are of particular concern as adult breeder or adult non-breeder), a social (high or low social they evolve rapidly, yielding new strains and adaptations to novel 3,5 status) and an infection (susceptible, infected or recovered) state. hosts . We fitted a multi-event CMR (MECMR) model that accounted We know little about the long-term demographic responses to for uncertainty on the assignment of infection states, i.e., we infectious viral disease epidemics in wildlife species, particularly included animals with unknown infection states10. We focused on those with high maternal investment in a low number of offspring three periods: pre-epidemic (pre-epidem; 1990–1992), epidemic during a long lifespan (K-selected species), probably because – 6 (epidem; 1993 1994), and post-epidemic (post-epidem; longitudinal studies on such species are rare . To characterise 1995–1999). The virulent CDV strain was not detected in any of these demographic responses, we apply two terms from the field 10 — our study clans after 1997 and it was not detected in any host in of ecology ecological resistance: the impact of exogenous dis- the Serengeti ecosystem after 199919. turbance on the state of a system; and recovery: the endogenous The annual apparent survival probability (ϕ) varied over time, process that pulls the disturbed system back towards equili- being highest during pre-epidem and lowest during epidem for brium7. In social host species, the relationships between social 8–10 all states, as the effect of periods was additive to the effect of traits and disease outcomes can be complex . It is therefore given states (Table 1). The transition probability from the unclear how social traits modulate the magnitude of the potential susceptible to the infected state (β, the infection probability) also reduction in host population size during an epidemic, or the 11 varied across periods; it was highest during epidem and lowest recovery time to pre-epidemic population size . during pre-epidem. CDV infection requires contact with aerosol There is a strong need for a robust predictive framework to droplets and body fluids from virus shedding individuals20. assess the potential risk that epidemics pose to wildlife popula- High-ranking subadults, breeders and non-breeders had a higher tions, to provide projections of the recovery of populations from infection probability than low-ranking ones (Table 1), a epidemics and to identify factors modulating population consequence of their higher contact rates10,21. As described responses, particularly for K-selected and social species. Here we 10,20 12,13 previously , CDV infection decreased survival only among show that stage-structured matrix population models are cubs and subadults but not adults (whether breeders or non- highly suitable for these purposes because they allow us to assess breeders, Table 1). Cubs of low-ranking mothers had a higher how demographic performance (in terms of fecundity, survival fl infection probability and were less likely to survive CDV and reproductive value) is in uenced by infection status, and how infection than those of high-ranking mothers. This is thought disease persistence and dynamics are influenced by host demo- 14,15 to be a consequence of their poorer body condition and lower graphy and social structure . Such models can make full use of allocation of resources to immune processes10. High-ranking field data as input, i.e., state-specific parameters estimated by ψ 16 females were more likely to become breeders ( , the probability capture-mark-recapture (CMR) approaches , and permit the of assuming a breeder state) than low-ranking females. High- calculation of the basic reproduction number (R0)ofa 14,15 ranking females were less likely than low-ranking ones to pathogen . R0 is a measure of the number of individuals maintain their social status (r, the probability of maintaining the infected by introducing a single infected individual into a sus- 17 current social state). Estimates of all parameters for each period, ceptible population (over the course of its infectious period) , which were used as input for the subsequent stage-structured i.e., a measure of maximum transmissibility, and describes whe- matrix population model, are shown in Table 1. Probabilities of ther an infectious disease will invade or fade out in a population. detection and assignment of infection states were assumed to be Here, we applied a stage-structured matrix population model constant throughout the study period. to a longitudinal dataset of 625 individually known female spotted hyenas Crocuta crocuta (hereafter hyenas) in the Seren- Population and CDV dynamics. The population growth rate (λ) geti National Park, Tanzania, spanning two decades. We inves- λ tigated the population consequences of a canine distemper virus differed substantially between periods (Fig. 1). was highest (CDV) epidemic10,18, caused by a novel genotype better adapted during pre-epidem and lowest during epidem. The hyena popu- to non-canids such as hyenas and lions Panthera leo than to lation increased during pre-epidem but decreased during epidem canids19. By quantifying and conducting a sensitivity analysis for and post-epidem (Fig. 1). fi We estimated CDV R0 in a closed hyena population, based on R0 of this CDV strain for the rst time in hyenas, we formally demonstrate that it was highly contagious among hyenas during our three study clans, and hence did not consider the spread of fl the non-canid strain in other non-canid species, such as the lion, the epidemic and that the most in uential parameter to decrease 19 R was the probability of becoming a socially high-ranking which were infected with this strain during the epidemic .We 0 modified a previously developed approach for discrete time breeder. CDV infection during the epidemic decreased the sur- 14,15 vival of cubs and subadults, particularly low-ranking ones. The models . Incorporating population growth rate into the projected hyena population showed a slow recovery, taking at calculation produced a measure of change in the rate of spread least 16 years to return to pre-epidemic levels. Population growth of the disease accounting for change in population size, i.e., measuring change in the incidence rate of the disease. We then rate and R0 were particularly sensitive to changes in the demo- fi graphic contribution of high-ranking females, in terms of their estimated this modi ed version of R0 during the epidemic period. probabilities of recruitment, survival and maintenance of high R0 was relatively high, equal to 5.69 ± 0.67, illustrating how CDV rank. Thus, high-ranking females increased the ecological resis- infection spread rapidly among hyenas during epidem. tance of the hyena population and helped to accelerate its recovery, thereby favouring the fadeout of the CDV epidemic. To Sensitivity analyses of λ and R0. To determine which vital rates fi λ our knowledge, this is the rst study in a wildlife host to (parameters) contributed most to changes in and R0, we con- λ demonstrate that the interaction of host demographic ducted sensitivity analyses for during each period and for R0

2 COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1 ARTICLE

Table 1 Input parameter values for the matrix population model

Process Parameter and notation m.l.e ± s.e.m Pre-epidem Epidem Post-epidem

Survival H susceptible cubs—ϕCHS 0.90 ± 0.04 0.84 ± 0.07 0.86 ± 0.06 L susceptible cubs—ϕCLS 0.86 ± 0.07 0.78 ± 0.10 0.81 ± 0.08 H infected cubs—ϕCHI 0.79 ± 0.06 0.67 ± 0.07 0.72 ± 0.06 L infected cubs—ϕCLI 0.62 ± 0.08 0.48 ± 0.08 0.54 ± 0.08 H susceptible subadults—ϕSAHS 0.97 ± 0.04 0.95 ± 0.06 0.96 ± 0.05 L susceptible subadults—ϕSALS 0.90 ± 0.12 0.83 ± 0.19 0.86 ± 0.16 H and L infected and recovered subadults—ϕSAIR 0.69 ± 0.04 0.56 ± 0.05 0.61 ± 0.03 H and L non-breeders—ϕNB 0.86 ± 0.02 0.78 ± 0.03 0.82 ± 0.02 H and L breeders—ϕB 0.95 ± 0.01 0.91 ± 0.02 0.93 ± 0.01 Infection H cubs—βCH 0.23 ± 0.09 0.96 ± 0.02 0.67 ± 0.16 L cubs—βCL 0.45 ± 0.16 0.99 ± 0.01 0.85 ± 0.10 H subadults, non-breeders and breeders—βH 0.10 ± 0.04 0.90 ± 0.05 0.42 ± 0.16 L subadults, non-breeders and breeders—βL 0.02 ± 0.01 0.66 ± 0.12 0.13 ± 0.08 Demography H subadults becoming H breeders—ψSAH 0.04 ± 0.02 L subadults becoming L breeders—ψSAL 0.01 ± 0.01 H non-breeders becoming H breeders—ψNBH 0.68 ± 0.02 L non-breeders becoming L breeders—ψNBL 0.60 ± 0.03 H breeders remaining H breeders—ψBH 0.49 ± 0.02 L breeders remaining L breeders—ψBL 0.45 ± 0.03 Social H remaining H—rH 0.94 ± 0.01 L remaining L—rL 0.97 ± 0.00 Other Average sex ratio in the population—sex.ratio 0.52 Average litter size in the population—litter.size 1.53

Demographic states were abbreviated as C cubs, SA subadults, B adult breeders, or NB adult non-breeders, social states as H high status or L low status and infection states as S susceptible, I infected or R recovered. Note that the parameter values for demography and social were calculated across all three periods (pre-epidem, epidem and post-epidem) Maximum likelihood estimate and associated standard error (m.l.e ± s.e.m.) of the probabilities of surviving, becoming infected, becoming a breeder, and maintaining the current social state in female hyenas during each period, as estimated via the multi-event capture-mark-recapture (MECMR) model. Sex ratio and litter size were estimated differently

and non-breeders were allocated identical probabilities of 1.2 assuming a breeder state, then the survival of breeders had a higher influence on λ than that of non-breeders. fl 1.1 The three most in uential parameters to decrease R0 during epidem were increases in the probabilities that high-ranking breeders remained and high-ranking non-breeders became high- 1.0 ranking breeders, and an increase in the probability of staying high-ranking across all demographic and infection states fl 0.9 (Fig. 3). The most in uential parameters to boost R0 were

Population growth rate growth Population increases in the probabilities of becoming infected; the probability of staying low-ranking; and the survival of susceptible cubs and 0.8 subadults (Fig. 3). As the infection probability was very high Pre-epidem Epidem Post-epidem during epidem (Table 1), a high survival of susceptible cubs and subadults indicated a high infection risk. For each parameter λ Fig. 1 Population growth rate ( ) of female spotted hyenas. Data shown are except the survival of susceptible cubs, high-ranking females for female spotted hyenas in the Serengeti National Park during the three contributed substantially more than low-ranking ones to either an – – epidemic periods (pre-epidem: 1990 1992; epidem: 1993 1994; post- increase or decrease in R epidem: 1995–1999). Empty circles represent outliers, the boxes 0. encompass the first to the third quartiles, inside the box the thick horizontal line shows the median and the whiskers are located at 1.5 × IQR Stable stage distribution (asymptotic values). The proportions (interquartile range) below the first quartile and at 1.5 × IQR above the third of female cubs, subadults, breeders and non-breeders quartile varied little across periods (Supplementary Fig. 1). On average (±s.d.), breeders (0.35 ± 0.04) and non-breeders (0.34 ± 0.04) were more common than cubs (0.18 ± 0.03) or subadults (0.12 ± during epidem. For λ, during all periods, the two most influential 0.02). parameters were the probabilities that high-ranking breeders and The proportions of low and high social states were similar and non-breeders stayed or became high-ranking breeders (Fig. 2). In varied little across periods (Supplementary Fig. 2), with slightly all periods, the probability of staying a high-ranking female had more low-ranking females (0.54 ± 0.04) than high-ranking ones an important positive contribution on λ (Fig. 2). For all para- (0.46 ± 0.03). The higher proportion of low-ranking females was a meters and during each period, high-ranking females contributed consequence of a higher probability of transition from a high to a substantially more to λ than low-ranking ones. The survival of low social state than from a low to a high social state (see non-breeders had a higher contribution than that of breeders Table 1). because non-breeders were more likely to assume a breeder state The proportions of susceptible, infected and recovered females than breeders at the next projection interval (Table 1). If breeders substantially varied across periods (Fig. 4 and Supplementary

COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio 3 ARTICLE COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1

a High-ranking 1.0 Low-ranking

0.5 

0.0 Sensitivity Sensitivity

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      BH NBH SAH rH NBHR BHR SAL BL rL NBL NBLR

b 1.0

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      BH NBH NBHR rH BHR SAH rL BL NBL NBLR SAH BLR

c 1.0

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      BH NBH NBHR rH SAH BHR rL BL NBL SAH NBLR BLR

Fig. 2 Sensitivity of the population growth rate (λ) to changes in vital rates in female hyenas during each period. a pre-epidemic, b epidemic and c post- epidemic. We only show the vital rates with effects on λ exceeding 10%, and order them according to their decreasing contribution to the absolute value of change of λ. Empty circles represent outliers, the boxes encompass the first to the third quartiles, inside the box the thick horizontal line shows the median and the whiskers are located at 1.5 × IQR (interquartile range) below the first quartile and at 1.5 × IQR above the third quartile. The pink boxes represent the vital rates for high-ranking females and the yellow boxes those for low-ranking ones. For the notation of parameters, see Table 1. Note that some parameters, such as the survival of high-ranking recovered non-breeder females ϕNBHR,were not present as input parameter values in Table 1. This is because the function used to conduct the sensitivity analysis of λ considered each parameter of the symbolic matrix as unique in itself, even if input values were similar

Fig. 3). Asymptotic, predicted values as expected from a stable and post-epidem, the pool of susceptibles was rapidly depeleted, stage distribution were as follows: during pre-epidem, the owing to the very high probabilities of infection (Table 1). proportion of susceptible females was high (0.60 ± 0.02) and the proportions of infected and recovered were low (0.08 ± 0.01 and 0.31 ± 0.03, respectively). During epidem, the proportion of Reproductive values. As is typical for the reproductive value, susceptible individuals was very low (0.01 ± 0.001), whereas the representing the expected current and future reproductive output22, relative proportions of infected and recovered individuals was it initially increased with age, that is, from cubs to subadults and high (0.17 ± 0.002 and 0.82 ± 0.07, respectively). During post- also from subadults to breeders and non-breeders. Across study epidem, there was a notable increase in the proportion of periods, the reproductive value of subadults was 2.2 times higher susceptible and a decrease in the proportion of recovered than that of cubs and the reproductive value of adults (pooled individuals (susceptible: 0.15 ± 0.01, infected: 0.16 ± 0.001, recov- breeders and non-breeders) was 6.2 and 2.8 times higher than that ered: 0.70 ± 0.06). of cubs and subadults, respectively (Supplementary Fig. 4). Among adults, the reproductive value of breeders was higher than that of Transient dynamics. Figure 4 shows how the proportions of non-breeders across periods (averaged values across periods: females in each infection state changed over the first 10 projected breeders: 17.4 ± 0.11, non-breeders 14.1 ± 0.16). Reproductive years and then reached a stable stage distribution. During epidem values varied little across periods (Supplementary Fig. 4).

4 COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1 ARTICLE

High-ranking Low-ranking 4 Both/other

2

0 0 R

Sensitivity –2

–4

–6

          BH H L NBH r H NB SAH CH CL r L BL B NBL CLS CHS SAHS SAL SALS sr SAIR

Fig. 3 Sensitivity of R0 to changes in vital rates in female hyenas during the epidemic period. R0 is the basic reproduction number for the virulent 1993/1994 CDV strain in hyenas. We only show the vital rates with effects on R0 exceeding 10% and order them according to their decreasing contribution to the change in absolute value of R0. Empty circles represent outliers, the boxes encompass the first to the third quartiles, inside the box the thick horizontal line shows the median and the whiskers are located at 1.5 × IQR (interquartile range) below the first quartile and at 1.5 × IQR above the third quartile. As in Fig. 2, pink boxes represent the vital rates for high-ranking females and yellow boxes those for low-ranking ones. The grey boxes represent vital rates for both high and low-ranking combined or vital rates unrelated to rank. For the notation of parameters, see Table 1. Note that sr here represents sex ratio

S I a 1.0 b 1.0 c 1.0 R

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4 Proportion Proportion Proportion 0.2 0.2 0.2

0.0 0.0 0.0 0246810 0246810 0246810 Years Years Years

Fig. 4 Dynamic projections of the proportion of different infection states and their convergence to a stable stage distribution. Projections are shown for medium term (first 10 years). We show dynamic projections for susceptible [S] (blue), infected [I] (orange) and recovered [R] (green) females (across all demographic and social states) during a pre-epidem, b epidem, c post-epidem. The starting values in each panel correspond to the (time invariant) initial state vector projection from the MECMR model. This figure illustrates which infection state predominates in each epidemic period, and how quickly each state converges to the stable stage distribution

High-ranking females, irrespective of their demographic state three periods onto the period from 2000 and 2010, when the or infection state, had a reproductive value (31.6 ± 1.69) almost virulent strain was absent from the ecosystem (Fig. 5, pink curve twice as large as that of low-ranking females (16.3 ± 1.22) across —see Supplementary Fig. 7 and Supplementary Results for an all periods, demographic states and infection states (Supplemen- alternative model). In addition, we predicted changes in popu- tary Fig. 5). lation size between 2010 and 2020, based on the parameters Across all periods, reproductive values of females were similar estimated for the period 2000–2010. Ecological resistance, the between different infection states (susceptible: 17.1 ± 1.46, opposite of the magnitude of the reduction (16%) in population infected: 16.5 ± 1.44, recovered: 14.3 ± 1.54, Supplementary Fig. 6). size, was 84% and recovery time, the time needed for the popu- The lower value for recovered individuals, as compared to lation to regain its pre-epidemic size, was 16 years. susceptible and infected ones, resulted from the fact that the We fitted a MECMR model without any contribution of social recovered state did not include any cub (because cubs were never status to all processes to show how population size would vary diagnosed as recovered in our original data set10), whereas both between 1990 and 2020 if all parameter values would be similar the susceptible and the infected states included cubs. for high-ranking and low-ranking females (Fig. 5, blue curve). By doing so, we highlight the role of high-ranking females in Ecological resistance and recovery. We projected changes in increasing ecological resistance and accelerating population population size using the values of λ obtained for each of the recovery after the epidemic. For this model, the drop in

COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio 5 ARTICLE COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1

Model Full model Model without social structure

160

120 Abundance

80

1990 1994 1998 2002 2006 2010 2014 2018 Year

Fig. 5 Mean abundance of female hyenas. Mean abundance (±95% confidence intervals) of female hyenas projected throughout the study period (1990–2010) and predicted beyond (2010–2020) based on the full model (pink) and a model without social structure (blue). The vertical bar (light orange) represents the period (1993–1994) during which the CDV epidemic occurred. This figure does not illustrate the actual number of hyenas in the study population; the population growth rate estimates from each period were used to produce it. The starting abundance was indexed as 100

population size was 23%, ecological resistance was therefore 77 % because subclinical CDV infection in adult females, if present, did and full recovery was not reached by 2020 (Fig. 5). not reduce their survival or chance of becoming a breeder (see Table 1 and ref. 10), and possibly also because of the rapid transition to the recovered state. Had infection negatively Discussion impacted female survival or fecundity, we would have expected The predicted impact of the CDV epidemic on female hyenas was larger differences in female reproductive values between the severe. The projected hyena population was reduced by 16% and susceptible, infected and recovered states. Interestingly, CDV R0 needed more than a decade to return to its pre-epidemic size. decreased as the probability for high-ranking females to maintain CDV is a global multi-host pathogen, infecting and causing their social status increased, probably because high-ranking mortality in a wide range of hosts, mostly carnivores23. In Africa, females were more likely to produce a litter than low-ranking CDV is considered to be an exotic pathogen imported from the females, and their cubs were less likely to become infected and to USA to South Africa by human activity in the previous century24. die of CDV infection (Table 1). We detected an elevated demo- As expected from other morbilliviruses, such as measles or graphic contribution by high-ranking females to population phocine distemper virus, R0 in hyenas was elevated during the growth (Fig. 2). The importance of high social status for popu- epidemic, demonstrating the capacity of the novel, virulent strain lation growth is further illustrated by the predicted persistent first observed during this epidemic to invade its new host population reduction and lack of full recovery when we modelled population in 1993. Here we estimated a CDV R0 in a closed the female population as homogeneous in terms of social states hyena population, not considering contacts between hyenas and (Fig. 5, blue curve). lions or other carnivores present in the Serengeti ecosystem. We In contrast to the lion population, which was reduced by 30% 18 consider it unlikely that our estimate of CDV R0 in hyenas was during the same epidemic , hyenas were relatively ecologically substantially affected by other carnivores since CDV transmission resistant with a potential reduction of only 16%. This difference occurs primarily via physical contacts, the frequency of contacts could be explained by the fact that in hyenas CDV infection only among hyenas (which are highly social) is substantially higher decreased juvenile survival (Table 1), whereas in lions mortality than between hyenas and other carnivores, and infectiousness was occurred in all age classes18. As the hyena population growth rate mostly among juveniles stationed at communal dens, who have was highly sensitive to adult survival (Fig. 2), the ecological limited contact to other non-canid species shedding CDV20,25. resistance of hyenas would have been far lower if infection had Adults allocated to the infected state were likely not infectious decreased adult survival. themselves, i.e. not actively shedding the virus10,19,20. Nikolin Although survival of adult hyenas was not affected by CDV, et al.19 provide a more detailed discussion of the likely involve- our projections show that hyenas needed potentially more time to ment of various carnivore species from an ecosystem-wide recover from this epidemic (~16 years, Fig. 5) than lions (~3 perspective. years)26. The most likely reasons for such a difference between An interesting finding was that reproductive values varied little the two species are different life histories and maternal invest- with infection states (Supplementary Fig. 6). This is probably ment patterns. Lionesses have no dominance hierarchy, produce

6 COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1 ARTICLE litters of 1–6 cubs, nurse communally and share food27. In con- The states of the matrix population model. We only modelled the female section trast, female hyenas have a linear dominance hierarchy and of the hyena population, as typical for demography models, which often focus on 22 compete intensively for food, produce smaller litters of 1–3 cubs, the philopatric sex . The four demographic states for female hyenas in the stage- structured matrix population model were as follows: cub (C), subadult (SA), and do not nurse communally. Instead, hyenas provide excep- breeder (B) and non-breeder (NB) (Table 2). Cubs were younger than 1 year, tionally rich milk and high maternal input in terms of individual subadults aged between 1 and 2 years. Cubs entirely depended on maternal milk for lactation effort which can last for 12–20 months28,29 and intense at least the first 6 months, subadults at least partially, as weaning occurs at – 28,29 within-litter competition can result in a reduction in litter size30. 12 20 months of age . Breeders gave birth to a litter during a given year, as documented by a freshly ruptured clitoris caused by parturition35 and/or sub- During the epidemic, hyena cub mortality was also spread across sequent lactation, whereas non-breeders did not. As breeders were mostly lactating several cohorts born between 1993 and 1994, increasing the time females, considering this state allowed us to account for the elevated energetic cost required for the population to compensate increased cub mor- of lactation and the possible delayed and indirect costs of females losing their tality. Our results thus suggest that for K-selected species with a offspring during the outbreak30. The two social states were low (L) or high social status (H) (Table 2). These slow development of young, high resilience may depend on high social states were based on the positions of females in strictly linear adult female ecological resistance to the initial epidemic rather than a fast dominance hierarchies in study clans. To construct these hierarchies we recorded, recovery if pathogen infection is virulent to juveniles but benign for each clan, submissive behaviours during dyadic interactions between adult – to adults. In addition, in social species with dominance hier- females30,36 38. Hierarchies were updated after recruitment or loss of adult females and after periods of social instability. To compare the social status of each female archies, it is likely that high-ranking animals will substantially (across clans and years), females were assigned standardised ranks30,36–38. contribute to population recovery after an outbreak. Standardised ranks were evenly distributed between the highest (+1) and lowest Epidemics of emerging infectious diseases pose a threat to (−1) standardised rank. Adult (i.e. breeder and non-breeder) females were many K-selected species. For example, Ebola outbreaks recently classified as holding a high status if their standardised rank was between 0.01 and + − 10 caused a severe decline in a western lowland gorilla population; 1 or a low status if their standardised rank was between 1 and 0.0 . Every year, 11 we assigned half of the females with a standardised rank above the median to the this population has yet to recover . Substantial advances in our high social status H and half of the female with a standardised rank below the understanding of infectious diseases and ability to forecast their median standardised rank to low social status L10. We assigned the social state of outcome has been considerably improved by the development of the genetic mother to non-adopted cubs and subadults, and that of the surrogate 31,32 mother to adopted ones10 (as offspring typically acquire a rank immediately below continuous time models . However, the theoretical develop- 38 33 that of the female that reared them ). High-ranking females have preferential ment of these models has outpaced their fusion with data , partly access to food resources in clan territories and commute less frequently to forage in 34 owing to the practical difficulty of estimating their parameters . areas outside the territory than low-status females. As a result, they produce their To accurately predict and improve the diagnosis of the impact of first litter at an earlier age37 and produce more milk, which results in their cubs future epidemics on vulnerable wildlife populations, it is thus growing faster and surviving better to adulthood than the cubs of low-ranking females37. We found previously that the cubs of high status females have a higher essential that we develop appropriate data-driven models. probability of surviving infection with CDV (Table 1) than those of low-ranking Because discrete time modelling approaches can easily incorpo- females10. rate (typically uncertain) field data and because they allow the The three infection states for female hyenas were susceptible (S), infected (I) interactions between host demography, sociality and disease and recovered (R) (Table 2). The outcomes of diagnostic procedures were employed to assign these states; (1) RT-PCR screening for the presence or absence dynamics to be assessed, we expect their use to increase in disease of CDV RNA in samples, (2) CDV antibody titres in serum and (3) the observation ecology. By monitoring changes in the stable state proportions of of clinical signs associated with CDV infection in hyenas, and the secondary individuals in terms of susceptible, infected and recovered (Sup- infections it causes in this species20 (for more details on these procedures, plementary Fig. 3) through time, discrete time modelling including on antibodies used and sample sizes, see the Supplementary Methods and approaches allow to predict when a population is at risk following Supplementary Table 1). CDV infection could occur only once in life because hyenas develop life-long immunity to this virus if they survive the infection10,23.As the emergence of a virulent strain. This would be expected when a a result, any individual hyena classified as susceptible in a given year was further higher proportion of susceptible individuals than recovered ones classified as susceptible during all previous years when the individual was detected; occurs. These types of models can also be useful to compare the infected in a given year was classified as susceptible during all previous years, and outcomes of management interventions by predicting their recovered during all subsequent years following the infected state; recovered in a given year was classified as recovered during all subsequent years when the impacts on changes in population size and assessing which vital individual was detected10. 14,33 rates have the greatest impact on population viability . One SIR models generally assume that (all) infected animals are infectious, but this is interesting development of our deterministic model, which not the case for CDV39 and some other pathogens such as herpes viruses40. assumes constant environmental effects and constant survival Animals infected with CDV start shedding virus (i.e., become infectious) when epithelial cells are infected, which marks the onset of clinical Distemper39. The rates during each epidemic period, would be to include the effects initial stage of CDV infection in blood immune cells induces the production of of stochastic forces (environment or demography) or density- antibodies against CDV, but at this stage virus is not shed and infection can be dependent or frequency-dependent processes on the dynamics of cleared before Distemper develops23. As a result, not all hyenas infected with CDV disease transmission14,15. were infectious and Distemper was overwhelmingly observed in young hyenas10,19,20. Our MECMR model10 included individuals with an unknown infection state, that is, we modelled uncertainty on the infection state (Supplementary Methods). Methods As we focused on uncertainty related to infection states, we did not include a small MECMR model. In a previous study, we developed and validated a MECMR model fraction of individuals with unknown or unclear demographic and social states. to quantify yearly survival and transition probabilities between demographic, social Including such individuals would imply to account also for uncertainty on the and infection states in three hyena clans monitored for two decades in the demographic and social states and this would result in over-parameterised Serengeti National Park that were infected with CDV during the disease MECMR models. Excluding this small fraction of individuals is unlikely to have epidemic of 1993/199410. Here, we adapted this MECMR model to include tem- biased the parameters estimated in this study because MECMR models account for 16 poral effects on survival and infection probabilities, and used the parameter esti- potential heterogeneity in detection probability and sample sizes remained large 10 mates of this MECMR model (Table 1) as input for the stage-structured matrix even after exclusion of this fraction . All females were highly detectable, regardless population model. We present all methods and sample sizes established in our of their demographic or social state, as we show in10 and in the Supplementary previous study10 in detail in the Supplementary Methods. In the next section, we Results. To fulfil a main assumption of CMR analyses on sampling design, we explain how we assigned individual hyenas to their demographic, social and focused exclusively on hyenas detected during our routine observations at clan infection state, as the states used in the MECMR model match with those used in communal dens, i.e. did not include hyenas that were detected opportunistically the matrix population model in the current study. All procedures in our previous away from clan communal den areas. study10 were performed in accordance with the requirements of the Tanzanian As we were interested in the key impact and the long-term perspective (20 authorities who issued appropriate research permits (1990-xxx-ER-90-130 to 2018- years) of the impact of the disease on population dynamics rather than a fine- 321-NA-90-130) and The Ethics Committee on Animal Welfare at the scaled, short-term one, the duration of the infected state in our model exceeded the Leibniz Institute for Zoo and Wildlife Research, Berlin, Germany (permit actual duration of the infectious period with CDV20. For the same reason, we did number: 2014-09-03). not include an exposed state, i.e., we did not model SEIR dynamics.

COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio 7 ARTICLE COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1

Table 2 Notation and definition of demographic, social and infection states in the stage-structured matrix population model

The 4 demographic states The 2 social states The 3 infection states Cub (C): 0 − 365 days Low social status (L): Susceptible (S): never infected with CDV Subadult (SA): 366 − 720 daysbelow median standardised social rank (low-ranking) Infected (I): infected with CDV Adult breeder (B): High social status (H): Recovered (R): immune to CDV have given birth to a litter (that year) equal or above median standardised social rank (high- Adult non-breeder (NB): ranking) have not given birth to a litter (that year)

These states were assigned to 625 female hyenas each year during 1990 and 2010. We considered uncertainty in the assignment of infection states. For further details on the methods used to assign these states see10 and Supplementary Methods

CDV prevalence and the proportion of infected females in each demographic state are shown in Supplementary Fig. 8 and Supplementary Fig. 9, respectively. Sample sizes for each combination of a given demographic, social and infection F state are provided in the Supplementary Table 2.

Structure of the meta-matrix . The overall stage-structured matrix population BB model, termed the meta-matrix M, was of the form: B CSABNB B SA 2 3 C ZZF Z C SA BNB NBB 6 7 ð Þ SA 6 SAC ZZZ7 1 SA NB M ¼ 6 7 C 4 5 SA B ZBSA BB BNB NB NB NB NB ZNBSA NBB NBNB

Each entry in M in Eq. (1) was a 6×6 submatrix that accounted for survival, Fig. 6 Population model. Structure of the overall matrix population model or social and infection processes also corresponding to the transition from a starting meta-matrix M in Eq. (1), with its 8 survival-transition submatrices demographic state (4 columns corresponding to the demographic state: cub (C), indicated in bold. Arrows represent the survival-transition probabilities of subadult (SA), breeder (B), non-breeder (NB) on the top of each column in Eq. (1)) to the following demographic state (4 rows corresponding to C, SA, B, NB, on the female hyenas, i.e. the transitions between the 4 demographic states left side of the matrix). Thus, M presented in Eq. (1) is a block-matrix notation of (shown as grey circles and with C cub, SA subadult, B breeder, NB non- total dimension 24×24. By combining the demographic, social and infection states, breeder), conditional on survival and considering potential changes in social the female part of the population at time t was represented by a vector n(t) with and infection states. The matrices were named according to the following 4×2×3= 24 components. Using the notation from Table 2, the 24 state combinations ranged from Cub— demographic state, with the starting demographic state as index. F was the Low social status—Susceptible to Non-breeder—High social status—Recovered, i.e. fertility matrix, accounting for cub production by breeders the infection state varied within the social state, itself varying within the demographic state, in the sequence detailed in Table 2. We used a discrete one-year states, Social a 6×6 submatrix accounting for transitions among social states and projection interval, as input parameter estimates were calculated on a yearly DemoBSA a 6×6 submatrix accounting for the demographic transitions, i.e. the basis10. The 24×24 meta-matrix M linked n(t)ton(t + 1) as: probability of becoming a breeder (recruitment) for subadults. The other submatrices SA ,NB ,B,NB and NB in M (Eq. (1), Fig. 6) nðÞ¼t þ 1 M ´ nðÞt ð2Þ C SA B B NB were built using the same logic, and were as follows: ¼ ´ ð Þ SAC SurvivalSA Infection 4 The notation Z in Eq. (1) was for a null 6×6 submatrix, indicating that there was no transition from the 6 columns corresponding to the 6 social × infection states in ¼ ´ ´ ´ ð Þ this demographic state to the corresponding 6 rows in this demographic state. For NBSA SurvivalNB Infection Social DemoNBSA 5 instance, the first entry in M (Eq. (1), top left) is a null 6×6 submatrix because there was no transition from the cub state to the cub state, as by definition all surviving ¼ ´ ´ ´ ð Þ cubs became subadults (Table 2). The notations SAC, BSA, NBSA, F, BB, NBB, BNB BB SurvivalB Infection Social DemoBB 6 and NBNB in Eq. (1) corresponded to 6×6 survival-transition submatrices, named according to the following demographic state, with the starting demographic state ¼ ´ ´ ´ ð Þ as index, using the state notations from Table 2. These submatrices accounted for NBB SurvivalNB Infection Social DemoNBB 7 demographic transitions for female hyenas, conditional on their survival and considering potential changes in social states and in infection states. For instance, ¼ ´ ´ ´ ð Þ SAC represented the survival-transition probabilities from the cub state to the BNB SurvivalB Infection Social DemoBNB 8 subadult state. The notation F in Eq. (1) corresponded to a 6×6 submatrix that modelled fertility. The overall matrix population model M can also be represented ¼ ´ ´ ´ ð Þ as a life cycle graph (Fig. 6). NBNB SurvivalNB Infection Social DemoNBNB 9

Assembling the submatrices. All processes (transitions between states and sur- vival) were first modelled separately from each other and then combined in a way For the 6×6 submatrix accounting for fertility F (Eq. (1), Fig. 6), we assumed described here. The 6×6 submatrices in M in Eq. (1) were obtained by appropriate that all births and deaths occurred simultaneously at the end of the projection products of parameters in separate matrices (described in the next section), to interval, that is, we modelled the population with a birth pulse reproduction and a generate successive events within a 1 year projection interval: change in demo- pre-breeding census22. Heterogeneity in the timing of births was handled in the graphic state, change in social state, change in infection state and then survival. For statistical analysis of our data. The fertility submatrix was: instance, the survival-transition submatrix to the breeder state from the subadult ¼ ´ ´ ´ ð Þ F SurvivalC InfectionC InfectionN fecundity 10 state (BSA in Eq. (1), Fig. 6) was expressed as: ¼ ´ ´ ´ ð Þ BSA SurvivalB Infection Social DemoBSA 3 In Eq. (10), SurvivalC was a 6×6 submatrix accounting for cub survival, InfectionC a 6×6 submatrix accounting for transitions between infection states for In Eq. (3), SurvivalB was a 6×6 submatrix accounting for the survival of cubs, InfectionN a 6×6 submatrix accounting for transitions between infection breeders, Infection a 6×6 submatrix accounting for transitions among infection states for newborns and fecundity, the elementary fecundity submatrix.

8 COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1 ARTICLE

The structure of such block-matrix model in M (Eq. (1)) was similar to the a  structure of the MECMR model used to estimate demographic, social and infection B b parameters, reflecting the transitions from one state to another on a yearly basis10. B B  Description of all submatrices. Each entry in the submatrices described below SA corresponded to the probability of transition (or survival), or the fecundity, from a  1 –  starting combination of social and infection states to a following combination of SA NB SA B 1 –  social and infection states. As indicated above, the 6 rows and 6 columns in these NB — —  submatrices corresponded to: Low social status Susceptible, Low social status 1 – SA Infected, Low social status—Recovered, High social status—Susceptible, High social NB NB status—Infected, High social status—Recovered. We built six diagonal demography submatrices; three to account for the transition to the breeder state (DemoBSA, DemoBB, DemoBNB) (Fig. 7a) and three Fig. 7 Transitions between demographic states. Transition to a the breeder others to account for the transition to the non-breeder state (DemoNB , SA state and b the non-breeder state, for female hyenas. The grey circles show DemoNBB, DemoNBNB) (Fig. 7b), both accessed from the subadult, breeder or non-breeder state. The parameter ψ in these submatrices denoted the transition the 3 demographic states from which the breeder and non-breeder states probability to the breeder state and 1−ψ the transition probability to the non- can be accessed (SA subadult, B breeder, NB non-breeder). Arrows and breeder state. As indicated above and in Table 2, all surviving cubs became fi symbols in italics represent the transition probabilities to a the breeder subadults by de nition, explaining why there is no demography submatrix in Eq. ψ ψ ψ (4). These matrices were as follows: state from subadults ( SA), breeders ( B) and non-breeders ( NB) and b to 2 3 −ψ −ψ À ψ the non-breeder state from subadults (1 SA), breeders (1 B) and non- L S SAL 0 0 0 0 0 −ψ 6 7 breeders (1 NB). All transition probabilities varied with social status (see L À I 6 0 ψSAL 0 0 0 0 7 6 7 Table 1 for parameter notations and estimates) − this is not shown, for À 6 ψ 7 L R 6 00SAL 0 0 0 7 DemoB ¼ 6 7 ð11Þ simplicity SA H À S 6 000ψSAH 0 0 7 6 7 H À I 4 0000ψSAH 0 5 H À R 0000 0ψSAH a r L r H 2 3 À ψ L S BL00000 1 - r H 6 7 L À I 6 0 ψBL00007 6 7 L H À 6 ψ 7 L R 6 00BL0007 DemoB ¼ 6 7 ð12Þ 1 - r L B H À S 6 000ψBH 0 0 7 6 7 À 4 ψ 5 H I 0000BH 0 b H À R 0000 0ψBH 1 -  1 2 3 L À S ψNBL 0 0 0 0 0 6 7 S I R L À I 6 0 ψNBL 0 0 0 0 7  6 7 1 À 6 ψ 7 L R 6 00NBL 0 0 0 7 DemoB ¼ 6 7 NB H À S 6 000ψNBH 0 0 7 6 7 Fig. 8 Transitions between social and infection states for female hyenas. 4 5 H À I 0000ψNBH 0 a Transitions between social states in subadult, breeder and non-breeder H À R 0000 0ψNBH female hyenas corresponded to the submatrix Social (Eq. (17)). The two ð13Þ social states are indicated as grey circles (L: low ranking; H: high-ranking). The arrows show the transition probabilities, with r and r the probabilities 2 3 L H L À S 1 À ψSAL 0 0 0 0 0 of staying in a low or high social state, respectively, and with 1−r and 1−r 6 7 L H L À I 6 01À ψSAL 0 0 0 0 7 6 7 the probabilities of becoming high or low-ranking, respectively. b À 6 À ψ 7 L R 6 001SAL 0 0 0 7 DemoNB ¼ 6 7 Transitions between infection states in subadult, breeder and non-breeder SA H À S 6 0001À ψSAH 0 0 7 6 7 female hyenas, corresponding to the submatrix (Eq. (20)). The H À I 4 00001À ψSAH 0 5 Infection H À R 0000 01À ψSAH three infection states are indicated as grey circles (S susceptible, I infected, ð14Þ R recovered). The arrows show the transition probability between those states, with β the probability of transition from a susceptible state to an 2 3 L À S 1 À ψBL 0 0 0 0 0 infected state (the infection probability). The infection probability varied 6 7 L À I 6 01À ψBL 0 0 0 0 7 with the social state (see Table 1) − this is not shown, for simplicity 6 7 À 6 À ψ 7 L R 6 001BL 0 0 0 7 DemoNB ¼ 6 7 B H À S 6 0001À ψBH 0 0 7 6 7 H À I 4 00001À ψBH 0 5 H À R 0000 01À ψBH ð15Þ becoming high or low ranking, respectively. Cubs had the same social state as their 38 2 3 genetic or surrogate mother ; which explains why this social submatrix does not L À S 1 À ψNBL 0 0 0 0 0 appear in Eq. (4). This submatrix was as follows: 6 7 L À I 6 01À ψNBL 0 0 0 0 7 2 3 6 7 À À À 6 À ψ 7 L S rL0 01rH0 0 L R 6 001NBL 0 0 0 7 DemoNB ¼ 6 7 6 7 NB H À S 6 0001À ψNBH 0 0 7 L À I 6 0 rL0 01À rH07 6 7 6 7 À 4 À ψ 5 À 6 À 7 H I 00001NBH 0 L R 6 00rL0 01rH 7 À À ψ Social ¼ 6 7 ð17Þ H R 0000 01NBH H À S 6 1 À rL0 0 rH0 07 ð16Þ 6 7 H À I 4 01À rL0 0 rH05 H À R 001À rL0 0 rH We built a social submatrix to account for the fact that subadults, breeders and non-breeders could either stay within their social state or change it (Fig. 8a). The parameters rL and rH in this submatrix represented the probabilities of staying in a We built three infection submatrices: one for newborns (InfectionN, Eq. (18)), − − low or high social state, respectively, and 1 rL and 1 rH the probabilities of one for cubs (InfectionC, Eq. (19)) and one for subadults, breeders and non-

COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio 9 ARTICLE COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1 breeders (Infection, Eq. (20), Fig. 8b). As mentioned above, we considered that follows: infected individuals became recovered at the following interval, conditional on 2 3 L À S 100000 their survival. We also considered that recovered individuals acquired life-long 6 7 L À I 6 0100007 immunity to CDV, which implied that they remained recovered until their death or 6 7 10 À 6 7 disappearance (as in ). For the infection submatrix for newborns InfectionN, part L R 6 0010007 of the submatrix F (in Eqs. (1) and (10), Fig. 6), we considered that mothers only fecundity ¼ sex:ratio ´ litter:size ´ 6 7 ð21Þ H À S 6 0001007 produced susceptible newborns, irrespective of their own infection status. Indeed, 6 7 À 4 5 as we assumed that adults were not infectious and hence did not excrete the virus H I 000010 actively as young individuals10,19,20, we considered the transmission of CDV to be H À R 000001 strictly horizontal. These key features of the model generated susceptible individuals in the model population at each projection interval (in addition to those already present in the initial population vector, details below). The 6 × 6 matrix for In Eq. (21), sex.ratio was the average sex ratio and litter.size the average litter newborns, InfectionN, was as follows: size in our study population. Each entry in fecundity was the elementary fecundity from a starting combination of social and infection states, to an ending 2 3 combination of social and infection states. L À S 111000 We built four submatrices to account for the survival of cubs, subadults, 6 7 L À I 6 0000007 breeders and non-breeders, respectively: SurvivalC, SurvivalSA, SurvivalB and 6 7 – À 6 7 SurvivalNB (Eqs. (22) (25)). The survival matrices were diagonal, representing the L R 6 0000007 ϕ Infection ¼ 6 7 ð18Þ survival probabilities (parameter ) of cubs, subadults, breeders and non-breeders N H À S 6 0001117 6 7 in given social and infection states, independently of transitions made during the H À I 4 0000005 projection interval. These matrices were as follows: 2 3 H À R 000000 L À S φCLS 0 0 0 0 0 6 7 L À I 6 0 φCLI 0 0 0 0 7 6 7 À 6 7 L R 6 0000007 Survival ¼ 6 7 ð22Þ C H À S 6 000φCHS 0 0 7 In Eq. (18), the value 1 represented the production of susceptible newborns by 6 7 À 4 φ 5 mothers of low social status and high social status (by definition all breeders H I 0000CHI 0 produced a litter thus the probability of female offspring production was equal to H À R 000000 1). For cubs, the only possible infection transitions were from a susceptible to a 2 3 L À S φSALS 0 0 0 0 0 susceptible state or from a susceptible to an infected state. The submatrix for cubs, 6 7 Infection (Eq. (19)) was thus: L À I 6 0 φSAIR 0 0 0 0 7 C 6 7 À 6 φ 7 L R 6 00SAIR 0 0 0 7 Survival ¼ 6 7 2 3 SA H À S 6 000φSAHS 0 0 7 L À S 1 À βCL00000 6 7 6 7 À 4 φ 5 L À I 6 βCL000007 H I 0000SAIR 0 6 7 À 6 7 H À R 00000φSAIR L R 6 0100007 Infection ¼ 6 7 ð19Þ ð Þ C H À S 6 0 001À βCH 0 0 7 23 6 7 H À I 4 000βCH 0 0 5 2 3 À φ À L S B0 0 0 0 0 H R 000010 6 7 L À I 6 0 φB0 0 0 07 6 7 À 6 φ 7 L R 6 00 B0 0 07 Survival ¼ 6 7 ð24Þ B H À S 6 000φB0 07 β β 6 7 In Eq. (19), CL and CH were the infection probabilities for low-ranking and H À I 4 0000φB05 high-ranking cubs, respectively, i.e. the probabilities of transition from a susceptible À φ −β −β H R 00000 B to an infected state (and hence 1 CL and 1 CH the probabilities of transition from a susceptible to a susceptible state for low-ranking and high-ranking cubs, 2 3 respectively). The values 1 corresponded to infected cubs becoming recovered at L À S φNB00000 6 7 the next interval (conditional on their survival). As there was no cub in the L À I 6 0 φNB00007 recovered state in the data set10, there was no transition from recovered cub to 6 7 À 6 φ 7 recovered cub. L R 6 00NB0007 Survival ¼ 6 7 ð25Þ The submatrix Infection accounted for possible transitions between susceptible, NB H À S 6 000φNB 0 0 7 β 6 7 infected and recovered states for subadults, breeders and non-breeders (Fig. 8b). L H À I 4 0000φNB 0 5 and β were the infection probabilities for low-ranking and high-ranking H À φ subadults, breeders and non-breeders, respectively, i.e. the probabilities of H R 00000NB transition from a susceptible to an infected state (and hence 1−βL and 1−βH the probabilities of transition from a susceptible to a susceptible state for low-ranking and high-ranking subadults, breeders and non-breeders, respectively). The values 1 As cubs were either susceptible or infected and never diagnosed as recovered in 10 in corresponded to infected individuals becoming or staying recovered at the next our original data set , in the last step of the model development we deleted the two interval (conditional on their survival) (Fig. 8b). This matrix was as follows: columns and two rows corresponding to recovered cubs. We thus ended up with a 22 × 22 meta-matrix M (Eq. (1)). 2 3 L À S 1 À βL00000 Asymptotic analyses of the matrix population model. We used R v. 3.5.0. (R 6 7 L À I 6 βL000007 Core Team 2017)41. We used the function ‘is.matrix_irreducible’ in R’s package 6 7 42 À 6 7 popdemo v 1.3-0 to verify that the meta-matrix was irreducible. Matrix models L R 6 0110007 Infection ¼ 6 7 ð20Þ are termed irreducible when their associated life cycles contain the transition rates H À S 6 0 001À βH007 6 7 to facilitate pathways from all states to all other states. Irreducible matrices are H À I 4 000βH005 ergodic: the stable asymptotic growth rate is independent from the initial stage structure in the population projection. Both conditions should ideally be met for H À R 000011 further analyses43. When M(t) = M is considered constant (i.e., no time dependence, no stochasticity), the behaviour of such a deterministic model is well-known22. The modelled population is then characterised by an asymptotic growth rate and a In accordance with the notation in the matrix M (Eq. (1), also see Fig. 6), young stable asymptotic distribution over the 22 states, which are the dominant females entered the population as cubs, born from breeders. As all females in the eigenvalue and the corresponding right eigenvector, respectively22. We used the breeder state reproduced by definition (Table 2), the fecundity value was set to 1. functions ‘stable.state’ and ‘reproductive.value’ in R’s package popbio 2.4.444 to Consequently, the elementary fecundity submatrix of newborn females per breeder determine the stable stage distribution and the reproductive values, based on the female (then combined with the other submatrices in the fertility matrix) was as definitions provided by22.

10 COMMUNICATIONS BIOLOGY | (2018) 1:201 | DOI: 10.1038/s42003-018-0197-1 | www.nature.com/commsbio COMMUNICATIONS BIOLOGY | DOI: 10.1038/s42003-018-0197-1 ARTICLE

‘ ’ ‘ ’ ’ We used the functions pop.projection and stage.vector.plot from R s package periods (1990–1992: pre-epidem; 1993–1994: epidem, 1995–1999: post-epidem) 44 popbio 2.4.4 to plot the short-term population dynamics and the convergence to and a further post-epidemic period (2000–2010) improved the model’s ranking, the stable stage distribution (Fig. 4), using the initial state probabilities estimated and used the parameter estimates for the first three periods (presented Table 1)as from the MECMR model (see Supplementary Results) as initial stage vector. The input for the stage-structured matrix population model. We assumed that the other fi rst function calculates the population growth rate and stable stage distribution by parameters (i.e., transition probabilities among demographic states and among repeated projections of the Eq. (2). social states) did not vary across periods. Including temporal effects on survival and fi 14 We included a modi cation in the approach developed by to estimate the infection probabilities improved the model’s ranking substantially (Supplementary basic reproduction number R0 of this CDV during the epidemic period, Results). As the best-ranked model from Marescot et al.10 produced some non- fi considering only hyena-to-hyena transmission (i.e. no interspeci c interactions as estimable parameters, we used the second best-ranked. fi ’ discussed). Our modi ed version of R0 accounted for the population s growth rate To obtain the model without effects of social status (Fig. 5, blue curve), we used and by doing this, we replaced the assumption of a stationary host population by the second best-ranked model and removed the formulation of the effect of social accounting for observed population dynamics and thus also for changes in the status on the relevant processes of (1) initial states, (2) survival, (3) infection, (4) proportion of infected individuals. As for equivalent time-continuous matrix breeding (demography), (5) social transition and (6) detection probabilities. 31 models , R0 is the dominant eigenvalue of the next generation matrix, itself being the product of the fundamental matrix by the reproductive matrix. The entries in the reproductive matrix in an epidemiological context correspond to the rate at Parameter estimates. All parameter estimates from the MECMR model used as in fi which new infections are produced by infected individuals. In our case, there is input for the matrix population model are provided Table 1 for the rst three only one infected state, thus only one type of infection is produced by only one type epidemic periods (pre-epidem, epidem, post-epidem). For survival and transition – of (infected) individuals. We detail our modified approach below. probabilities for the period 2000 2010, initial state, detection and assignment The starting point is the decomposition of the change in the number of infected probabilities, see Supplementary Results. in transmission and transitions: The sex-ratio at birth is balanced and was estimated in a previous study to be equal to 0.5246. Here we used this estimate as input value for the parameter sex. A ¼ T þ R ð26Þ ratio (Table 1, fecundity matrix, Eq. (21)). This parameter estimate had no standard error associated with it. We set an arbitrary standard error value equal to where A is the projection matrix, T is the transition matrix and R is the 0.03 for the sensitivity analyses. reproductive matrix. Under asymptotic conditions, the changes in one time step are In a previous study, we estimated average litter size based on observations within a population that grows at rate λ. To represent the change in the number of conducted in our three main study clans between 1987 and 200647 and the infected as a change in proportion relative to the overall population, one can monitoring of the fate of 1124 spotted hyena cubs from 735 litters, comprising 351 discount for population growth at rate λ and write the contribution matrix C to the (47.8%) singleton, 379 (51.6%) twin and five (0.7%) triplet litters. Thus, for the “ ” fi next generation of infected (i.e. our modi cation of R0) as: average litter size (parameter litter.size, Table 1, fecundity matrix, Eq. (21)) we used the value 1.53 (1124/735). This parameter estimate had no standard error À1 À1 À1 À1 À2 2 À1 À3 3þ C ¼ λ R þ λ Rλ T þ λ Rλ T þ λ Rλ T ¼ ð27Þ associated with it. We set an arbitrary standard error value equal to 0.03 for the sensitivity analyses. which reduces to: À À À À C ¼ λ 1RðI À λ 1TÞ 1 ¼ RðλI À TÞ 1 ð28Þ Code availability. The R programming code to replicate all our analyses and figures is available from Github: https://github.com/LucileMarescot/SIR-Matrix- model. It is divided in three main sections: (1) code 1_Model construction loads the input values for the three epidemic periods and builds the matrix model, (2) code Sensitivity analyses of λ, R0 and stochasticity. To determine which parameters 2_Model analysis_asymptotic presents the asymptotic analysis of the matrix λ contributed most to and R0 and predict the results of future changes in parameter model and (3) code 3_Model analysis_stochastic presents the stochastic analysis of estimates, we performed sensitivity analyses (Figs. 2 and 3). When elements of a the matrix model. Each section is fully annotated so that each step can be population matrix are composed of several vital rates, the classical first order replicated. sensitivity analysis is not recommended, as it does not allow one to disentangle the effects of demographic, social and infection parameters. We therefore conducted λ λ ‘ Data availability lower-level sensitivity analyses for and R0. For we applied the function vital- sens’ from the R package popbio 2.4.444. For R , we used a numerical approach in The original data from Marescot et al. (2018) supplied the parameter estimates 0 fi 48 which we varied each parameter by 0.1% while maintaining the others constant, used in this study and are archived in gshare : https://doi.org/10.6084/m9. figshare.5840970. The parameter estimates used in this study are provided in and calculated R0 at each iteration as previously described. To account for parameter uncertainty (Figs. 1, 2, 3 and 5), we used Monte Carlo Table 1 and in Supplementary Results. iterations. At each successive period (pre-epidem, epidem, post-epidem, plus the second post-epidemic period (2000–2010)), we simulated 1000 block-matrices M, Received: 10 January 2018 Accepted: 19 October 2018 implemented with the biological parameter estimates from the MECMR model (Table 1). First, we drew 1000 values from normal distributions with means equal to the regression coefficients of the MECMR model and with standard deviations equal to the standard errors associated with these regression coefficients. Sampling correlations among estimates were sufficiently small to be neglected. To obtain the biological parameter estimates and insure that they corresponded to probabilities bounded between 0 and 1, we back-transformed those simulated regression References coefficients using the logit-function after accounting for the structural interactions 1. 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Author contributions PLoS One 11, e0163548 (2016). H.H., J.D.L., S.K.S. and M.L.E. designed the initial research; M.L.E., H.H. and S.B. per- 26. Packer, C. et al. Viruses of the Serengeti: patterns of infection and mortality in formed or supervised the field and/or laboratory work to estimate the parameters; L.M. African lions. J. Anim. Ecol. 68, 1161–1178 (1999). and S.B conducted all statistical analyses and modelling procedures under the supervision 27. Hanby, J. P., Bygott, J. D., & Packer, C. in Serengeti II: Dynamics, of O.G., S.K.S, J.D.L and H.H., and S.B., L.M. and M.L.E. wrote the paper. All authors Management, And Conservation Of An Ecosystem, (eds Sinclair, A. R. E. & contributed critically to the drafts and gave final approval for publication. Arcese, P.) 315–331 (University of Chicago Press, Chicago, 1995). 28. Hofer, H. & East, M. L. in Serengeti II: Dynamics, Management, And Conservation Of An Ecosystem, (eds Sinclair, A. R. E. & Arcese, P.) 332–363 Additional information (University of Chicago Press, Chicago, 1995). Supplementary Information accompanies this paper at https://doi.org/10.1038/s42003- 29. Holekamp, K. E., Smale, L. & Szykman, M. Rank and reproduction in the 018-0197-1. female spotted hyaena. J. Reprod. Fertil. 108, 229–237 (1996). 30. Hofer, H., Benhaiem, S., Golla, W. & East, M. L. Trade-offs in lactation and Competing interests: The authors declare no competing interests. milk intake by competing siblings in a fluctuating environment. Behav. Ecol. 27, 1567–1578 (2016). Reprints and permission information is available online at http://npg.nature.com/ 31. Dieckmann, O. & Heesterbeek, J. Mathematical Epidemiology Of Infectious reprintsandpermissions/ Diseases: Model Building, Analysis And Interpretation. (Wiley, New York, 2000). 32. Gandon, S., Day, T., Metcalf, C. J. E. & Grenfell, B. T. Forecasting Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in epidemiological and evolutionary dynamics of infectious diseases. Trends Ecol. fi Evol. 31, 776–788 (2016). published maps and institutional af liations. 33. Hobbs, N. T. et al. State‐space modeling to support management of brucellosis in the Yellowstone bison population. Ecol. Monogr. 85, 525–556 (2015). 34. Clark, J. S. Models For Ecological Data: An Introduction. (Princeton University Open Access This article is licensed under a Creative Commons Press, Princeton, 2007). 35. Hofer, H. & East, M. L. The commuting system of Serengeti spotted hyaenas: Attribution 4.0 International License, which permits use, sharing, how a predator copes with migratory prey. III. Attendance and maternal care. adaptation, distribution and reproduction in any medium or format, as long as you give Anim. Behav. 46, 575–589 (1993). appropriate credit to the original author(s) and the source, provide a link to the Creative 36. East, M. L. & Hofer, H. Male spotted hyenas (Crocuta crocuta) queue for Commons license, and indicate if changes were made. The images or other third party ’ status in social groups dominated by females. Behav. Ecol. 12, 558–568 (2001). material in this article are included in the article s Creative Commons license, unless 37. Hofer, H. & East, M. L. Behavioral processes and costs of co-existence in indicated otherwise in a credit line to the material. If material is not included in the ’ female spotted hyenas: a life history perspective. Evol. Ecol. 17, 315–331 article s Creative Commons license and your intended use is not permitted by statutory (2003). regulation or exceeds the permitted use, you will need to obtain permission directly from 38. East, M. L. et al. Maternal effects on offspring social status in spotted hyenas. the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Behav. Ecol. 20, 478–483 (2009). licenses/by/4.0/. 39. Sawatsky, B., Wong, X.-X., Hinkelmann, S., Cattaneo, R. & von Messling, V. Canine distemper virus epithelial cell infection is required for clinical disease but not for immunosuppression. J. Virol. 86, 3658–3666 (2012). © The Author(s) 2018

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